All Questions
5,908 questions
9
votes
1
answer
782
views
Mean value property with fixed radius
Let $f$ be a continuous function defined on $\mathbb{R^n}$. It is well known that both the spherical mean value property (MVP) of $f$, i.e.
$$f(x)=\frac{1}{|\partial B(x,r)|}\int_{\partial B(x,r)}f,\ ...
CommunityBot
- 1
1
vote
1
answer
342
views
Singular conformally-Euclidean metrics
Suppose $W : \Bbb{R}^n \to \Bbb{R}_+$ is a continuous, positive function, with exactly $n$ zeros $\alpha_1,...,\alpha_n$. Define the following 'distance':
$$ d(\alpha_i,\alpha_j)=\inf\{\int_0^1 \sqrt{...
- 2,222
4
votes
0
answers
273
views
Real Analytic Function and nth Prime
It is trivial that there are no polynomial function $P$ with integer coefficients that has the property $P(n)=p_n$ where $p_n$ is the $n$th prime.While it is true that can always construct a smooth ...
- 153
1
vote
1
answer
978
views
Concentration bound for weakly dependent random variables
Hi,
Suppose we observe a sequence $R_1, ..., R_T$ of iid. random variables that equal $0$ with probability $p$ and with probability $1-p$ are sampled from a distribution with expected value $E(R) >...
- 37.7k
5
votes
1
answer
1k
views
Notions related to De Rham Cohomology
In R^2, we have the following abelian groups, some of which have R vector space structures, or even C vector space structures.
Closed forms/exact forms
real parts of analytic functions/harmonic ...
- 35.2k
4
votes
1
answer
314
views
Spectral Properties of $A(I-A)^{-1}$
I am working with a class of matrices $A$ which are non-negative-definite, not symmetric, and have maximum eigenvalue less than 1. I am interested in the spectral properties of the matrix $H = A(I - ...
- 20.2k
3
votes
0
answers
181
views
Example showing that area is discontinuous in the 2-variation seminorm
The $p$-variation seminorm (where $p \ge 1$) of a continuous curve $\alpha: [0,1] \to \mathbb{R}^2$ is defined as the supremum over all partitions $t_0 = 0 \le t_1 \le \cdots \le t_n = 1$ of:
$\left(\...
- 4,304
4
votes
2
answers
1k
views
$L^1$ norm of the Fourier transform of a truncated Gaussian
I asked this question on Math StackExchange recently but the only useful comment I got was that this could be a good question for Math Overflow. Here it goes:
Consider the Gaussian $G(x):=e^{-x^2}$ ...
- 981
0
votes
0
answers
345
views
Jacobian of the inversion map
Let $F:Tr(n,\mathbb{R})\cap GL_n(\mathbb{R})\rightarrow Tr(n,\mathbb{R})\cap GL_n(\mathbb{R})$ be the map which sends a matrix $A$ to its inverse $A^{-1}$. If we consider $F$ as a function from $(\...
- 657
4
votes
0
answers
109
views
rank of a C^1 map
I saw this three star problem in Hirsch ..
If we have open sets $U \subset R^3$ ,$V \subset R^2$ and $f:U \to V$ is $C^1$ and onto...Prove there is at least one point in $U$ where $f$ has full rank
...
- 15.2k
10
votes
2
answers
1k
views
Does Rolle's Theorem imply Dedekind completeness?
I think the answer to the title question is "yes", but Gerald Edgar, in his comment on Does antidifferentiability of continuous functions imply Dedekind completeness? , points out an article (actually ...
CommunityBot
- 1
6
votes
2
answers
4k
views
Is there dual space of the distributions $\mathcal{D}'(R)$?
Dear MOs,
Let $\mathcal{D}(R):=C_c^\infty(R)$ be the smooth functions with compact support. Its dual space is the space $\mathcal{D}'(R)$ of distributions. This space $\mathcal{D}(R)$ has its weak *-...
- 34.7k
10
votes
0
answers
315
views
Does antidifferentiability of continuous functions imply Dedekind completeness?
Let $R$ be an ordered field, and let $I$ be {$x \in R: a < x < b$} for some $a < b$ in $R$. Define notions of $R$-continuity and $R$-differentiability for functions $f : I \rightarrow R$ by ...
- 19.7k
1
vote
1
answer
254
views
references for families of conditionaly negative definite matrices
We say that a matrix $A\in M_n(\mathbb{C})$ is a conditionaly negative definite matrix if it is hermitian and if for all complex numbers $c_1,\ldots,c_n$ such that $c_1+\cdots +c_n=0$ we have
$$
\sum_{...
CommunityBot
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1
vote
1
answer
334
views
Property Sigma Algebra [closed]
Is the set { $ \cup_{i \in \mathbb{N}} C_{i} \times D_{i} : C_{i} \in \mathcal{L} \ , D_{i} \in \mathcal{B}^{n} \ $ } a sigma algebra on $\mathbb{R} \times \mathbb{R}^{n}$ ?
- 236k
0
votes
1
answer
1k
views
surjective function from non-measurable sets
let $V$ be the vitali set and let $g:V\to\mathbb R$ be a surjective function. then the fuction $f:\mathbb R\to\mathbb R$ such that $f(x)=g([x])$ will be a function that is surjective in any interval ...
- 60.5k
1
vote
2
answers
1k
views
Is there a periodic function without minimum period such that all the possible periods are irrationals? [closed]
Let $f:\mathbb R\to\mathbb R$ be a periodic function. We say $f$ is without minimum period if, $\forall t$ such that $f(x+t)=f(x)\forall x$, there is a $t'$ such that $0<t'<t$ and $f(x+t')=f(x)\...
- 23.2k
0
votes
1
answer
606
views
Difference between spaces of integrable functions w.r.t Lebesgue measure and Borel measure [closed]
Is there a difference between
$L^p(\mathbb R,\mathfrak B,\beta)$ and $L^p(\mathbb R,\mathfrak L,\lambda)$ ?
Here I denoted by $\lambda$ the Lebesgue measure, defined on the Lebesgue
$\sigma$-algebra $\...
- 9,853
5
votes
3
answers
5k
views
Zeros of "exponential" function
Define ${f}_{i}(x) = \sum_{j=1}^{i} (-1)^{i-j}{i \choose j}j^x$, where $i=1,2,3,...$ and $x \in \mathbb{R}$.
For integer $x \geq i$, ${f}_{i}(x)$ reduces to ${f}_{i}(x)=i!S(x,i)$, where $S(x,i)$ is ...
CommunityBot
- 1
0
votes
1
answer
316
views
Modulo dynamics on [0,1)
For $T: \mathbb{R} \mapsto \mathbb{{R}_{+}}$, we have $\{ {T}^{n}(\theta)\ mod \ 1\} \subset [0,1)$. (where ${T}^{n}(\theta)$ means applying $T$ $n$ times on $\theta$, not the $n$th power of $T(\...
- 3,343
0
votes
1
answer
659
views
Under what condition will this set contain a limit point of [0,1)?
Let $T_1,T_2,....T_n$ be numbers such that
$T_k= k$ no. of digits in decimal expansion of an irrational number, say $\alpha$, starting from $(\frac{k(k-1)}{2}+1)^{th}$ digit in the decimal expansion. ...
- 60.5k
2
votes
1
answer
1k
views
On an eigenvalue inequality
Let $\lambda_1 (\cdot)$ be the larger absolute value
eigenvalue of a $2\times2$ matrix and $\lambda_2 (\cdot)$
the smaller absolute value eigenvalue of a $2\times2$ matrix, i.e.
$|\lambda_1 (\cdot)| \...
CommunityBot
- 1
6
votes
2
answers
1k
views
On the uncountability of zero sets
If $f$ is any real-valued function, we define its zero set $Z_f = \{ x : f(x) = 0 \}$. Obviously, the zero set of a nice function can be uncountable. e.g., if $f(x) = 0$ on an uncountable domain.
I ...
- 8,512
1
vote
2
answers
450
views
A smoothness of $f(\sqrt[p] x)$
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a smooth function let $p \in \mathbb{N}$, $p \geq 2$.
Assume that $f^{(k)}(0)=0$ for all $k \notin p \mathbb{N}$. Is it true that then $g(x)=f(\sqrt[p] x)$...
5
votes
1
answer
543
views
Acceleration via smoothing
Is the following approach to accelerating the rate of convergence of $(1+1/2+\dots+1/n)- \ln n$ (with $n=1,2,3,\dots$), and other sequences like it, in the literature?
Let $f(t)=(\sum_{1 \leq n \leq ...
CommunityBot
- 1
3
votes
1
answer
500
views
Hausdorff measure on product spaces of p-adic integers
This question came up (unexpectedly) in a problem I was working on a few years ago. It may not be too difficult but I never got around to figuring out the answer, because all I needed at that time was ...
- 23.2k
6
votes
0
answers
8k
views
Dual space of continuous functions
Let $C_b(\Omega,V )=$ { $ f:\Omega\rightarrow V $ } is the Banach space of all bounded continuous functions in Banach space $V$ with a norm $\|\cdot\|$ defined as $\|f\|_\infty=\sup _{x\in\Omega}\|f(x)...
- 385
2
votes
1
answer
255
views
Quotients of perfect powers separated by an integer
Let $a_n=\frac{(n+1)^{n+2}}{n^n}$ and $b_n=\frac{(n+2)^{(n+1)}}{(n+1)^{n-1}}$. Then it is easy to see that $a_n \leq b_n$ for all integers $n\geq 1$ (because the sequence $(1+\frac{1}{n})^n$ is ...
- 1,723
4
votes
1
answer
627
views
Does such a smooth function exist?
I am looking for a $C^\infty $ function $g:\mathbb{R}^3\to \mathbb{R}^3$ such that $g(x)=0$ for $|x|\le 1$ and $g(x)=x$ for $|x|\ge 2$. Certainly such $g$ can be constructed, but I also want it to ...
CommunityBot
- 1
2
votes
1
answer
2k
views
Modified Lebesgue differentiation theorem
Let $\Omega\subset \mathbb{R}^n$ an open set and $u:\Omega\to \mathbb{R}$ be a (locally) $L^1$-function. Then it is well known that the Lebesgue differentiation theorem holds: For almost every $x\in \...
CommunityBot
- 1
21
votes
3
answers
3k
views
Prime ideals in the ring of germs of continuous functions
We all know that the ring of germs of continuous functions at a point on, say $\mathbb{R}$, has a unique maximal ideal- namely, those functions that vanish at that point.
Can anyone think of a single ...
- 47.5k
4
votes
2
answers
371
views
Heights of several interesting posets
Let the height of a poset $P$ be the supremum of ordinals that are order types of all well-ordered subsets of $P$ (with order inherited from $P$).
Define several sets of total functions, in each ...
- 2,943
9
votes
2
answers
1k
views
Fourier transform of x2 invariant measure
Let $T:\mathbb{R}/\mathbb{Z}\rightarrow \mathbb{R}/\mathbb{Z}$ be the map defined by $T(x)=2x$, and suppose that $\mu$ is a $T$ invariant and ergodic Borel probability measure on the space, which is ...
- 1,723
1
vote
2
answers
1k
views
An interesting doubly infinite series
Let $0<\mu<1$ and $\alpha:=1-\mu^2$. Consider the function
$$f(x):=x\sum_{k=-\infty}^\infty\mu^{4k}e^{-\alpha\mu^{4k}x}-\frac{1}{x}\sum_{k=-\infty}^\infty\mu^{4k}e^{-\alpha\mu^{4k}/x},$$
...
CommunityBot
- 1
0
votes
1
answer
130
views
Maximal length vector under constraints
Consider a criculant symmetric $M$ an $n \times n$ matrix with $0$ and $1$ entries and $r$ entries of $1$ in each row with the diagonal values taken as $1$. I am looking for a $0-1$ vector $v$ with ...
- 39.9k
2
votes
0
answers
495
views
Characterization of weak Lebesgue spaces [closed]
I would be interested to know whether the following is true:
Let $\Omega$ be a bounded open set in $\mathbf{R}^n$. Let $g$ be a nonnegative function $g : \Omega \to \mathbf{R}$. If there is a ...
- 373
2
votes
0
answers
917
views
Guessing game with guess cost
This is a question about Problem 328 on the website Project Euler. A description of the problem is provided in the previous link. I was wondering if there has been any research done on this question. ...
- 4,952
11
votes
4
answers
5k
views
The metric space associated to a measure space
Let $(X, \mathcal{A}, \mu)$ be a measure space such that $\mu(X) < \infty$. We say that two measurable sets $A$ and $B$ are equivalent if $\mu (A \Delta B) = 0$. The equation $$ d(A,B) = \mu (A \...
- 148k
0
votes
1
answer
224
views
Special functions on the unit disk
Let $\mathbb{D} = \{ (x,y) \in \mathbb{R}^2 \mid x^2 + y^2 < 1 \}$ be the unit disk.
We say a function $f : \mathbb{D} \rightarrow \mathbb{D}$ is a winner if it satisfies the following:
1) it is a ...
- 1,271
5
votes
1
answer
418
views
positive hermitian elements in $M_n(\mathbb{C})$
Elements of the set $P$ of positive hermitian $n×n$ matrices over complex numbers
have some special properties:
(i) they are closed under sum,
(ii) they are closed under multiplication by positive ...
- 1,887
1
vote
1
answer
6k
views
How to determine whether a multivariate function is bounded or not
Suppose there is a function $f:\mathbb{R}_+^n\mapsto \mathbb{R}$. Are there any systematic ways to determine whether the range of $f$ is bounded or not?
For example, there is a function $f(x,y)=-x^2+...
- 37.7k
1
vote
1
answer
420
views
density of a set
let $S=\{\sin (n)|n \in N\}$. We can prove $S$ is dense in $[-1,1]$. So is the set $\{\sin( n^2)|n \in N\}$; but the set $\{\sin (n^3)| n \in N\}$ is not dense in $[-1,1]$. How to prove this?
CommunityBot
- 1
6
votes
2
answers
929
views
reverse mathematics strength of "Lipschitz functions are somewhere differentiable"
What is the reverse mathematics strength of
"For all Lipschitz functions $\; f : \mathbb{R} \to \mathbb{R} \;$, $\;$ there exists a real number $x$ such that $f$ is differentiable at $x$." ?
(...
- 44.4k
2
votes
2
answers
711
views
Power function inequality
Let $x$ and $p$ be real numbers with $x \ge 1$ and $p \ge 2$ . Show that $(x - 1)(x + 1)^{p - 1} \ge x^p - 1$ .
I recently discovered this result. I am sure it is known, but it is new to me. It is ...
- 37.7k
9
votes
1
answer
10k
views
Can the supremum of continuous functions be discontinuous on a set of positive measure? [closed]
Given a sequence of continuous functions $f_n(x)$, all defined on a compact set $D$ and assuming $f_n(x)$ is uniformly bounded. Let $f(x) = sup_n f_n(x)$.
It is clear that $f(x)$ is not necessarily ...
- 1,546
1
vote
2
answers
382
views
A question about zeros of Tate type integral
Fix a positive integer $n$. Fix a continuous character $\chi$ of $\mathbb{R}^*$ with the form $\chi(x)=sign(x)|x|^t$ for some complex number $t$. If $\phi$ is a Schwartz function on $\mathbb{R}$, let $...
- 3,183
3
votes
0
answers
211
views
Elementary analysis: reference request
Given the continuous maps $[0,\infty) \to \mathbb R$ define the following "truncation at level $K$ operator", $T$:
$T(f)(t) = f(\min(t, S_f))$, where $S_f = \inf \{ s : f(s) \ge K \}$
So essentially ...
- 2,895
3
votes
3
answers
522
views
Closure of singular points
Let $f(x,y)$ be a complex degree $d$ polynomial that has this particular
form.
$$ f = \frac{f_{02}}{2} y^2 + \frac{f_{21}}{2} x^2 y +
\frac{f_{12}}{2} x y^2 + \frac{f_{03}}{6} y^3 + \frac{f_{40}}{...
- 1,811
5
votes
2
answers
560
views
implicit function theorem for algebraic sets
We know by the standard Implicit Function Theorem that
If $f:\mathbb R^4\rightarrow\mathbb
> R^2$ is a polynomial (or in fact any
continuously differentiable function),
then there is a ...
- 1,359
1
vote
1
answer
224
views
Can symmetrizing a contraction increase the speed of convergence?
Dear community,
I have a problem which is very simple to state but seems to be hard to answer.
Statement of the problem
Let $f$ and $g$ be two symmetric, real functions in $n$ and $m$ variables, ...
- 199